Complex numbers

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Complex numbers are numbers having both a real part and an imaginary part. A complex number can be written as $z \,=\, a + b\,i$ or $z \,=\, a + i\,b$ in Cartesian coordinates, with $a$ and $b$ being real numbers and $i$ being the imaginary unit " $\sqrt{-1}$ ", or as $z = r\, e^{i \theta} = r \, (\cos(\theta) + i \sin(\theta))$ in polar coordinates. Bernhard Riemann wrote complex numbers as $s = \sigma + i\,t$ . The set of complex numbers is denoted by $\mathbb{C}$ or C.

For example, $z = \Re(z) + \Im(z) i = \frac{1}{2} + (14.1347251417...) i$ is a complex number, which in polar coordinates is $z = | z | e i arg(z)$ , where $|z| = \sqrt{{\Re(z)}^2 + {\Im(z)}^2}$ (= 14.143565845691...) is the modulus of $z$ , and $\arg(z) = {\rm arctan2}(\Im(z), \Re(z), 0 \pi )$ (= 1.5354371953233...) is the argument of $z$ .

Complex conjugate and absolute value of a complex number

The complex conjugate $\scriptstyle \overline{z} \,$ of a complex number $\scriptstyle z \,=\, a + bi \,=\, r e^{i \theta} \,$ is defined as

$\overline{z} \,:=\, a - bi \,=\, r e^{-i \theta}. \,$

The absolute value $\scriptstyle |z| \,$ of a complex number $\scriptstyle z \,=\, a + bi \,=\, r e^{i \theta} \,$ is defined as

$|z| \,:=\, \sqrt{z \overline{z}} = \sqrt{(a + bi)(a - bi)} = \sqrt{a^2 + b^2} = \sqrt{r e^{i \theta} \cdot r e^{-i \theta}} = r. \,$

Considering complex numbers as points in the complex plane (Argand diagram), the absolute value of a complex number corresponds to the norm of the vector (a, b) in $\scriptstyle \mathbb{R}^2 \,$ (the real part $\scriptstyle a \,$ and imaginary part $\scriptstyle b i \,$ being "orthogonal").

Complex argument

The argument of a complex number $\scriptstyle z \,=\, a + bi \,=\, r e^{i \theta} \,$ is

$\arg(z) = \arg(a + b i) = \arctan\bigg(\frac{b}{a}\bigg) = \arctan\bigg(\frac{\sin(\theta)}{\cos(\theta)}\bigg) = \arctan(\tan(\theta)) = \theta, \,$

where $\scriptstyle \theta \,$ is considered modulo $\scriptstyle 2 \pi \,$ , i.e. $\scriptstyle \theta \,\in\, [0, 2 \pi) \,$ .

Real part and imaginary part of a complex number

The function $\scriptstyle \Re(z)\,$ gives the real part of a complex number $\scriptstyle z \,=\, a + bi \,=\, r e^{i \theta} \,$

$\Re(z) \,\equiv\, a = \frac{z + \overline{z}}{2} = r \cos(\theta). \,$

The function $\scriptstyle \Im(z)\,$ gives the imaginary part of a complex number $\scriptstyle z \,=\, a + bi \,=\, r e^{i \theta} \,$

$\Im(z) \,\equiv\, bi = \frac{z - \overline{z}}{2} = r \sin(\theta) ~ i.\,$

Thus

$z = \Re(z) + \Im(z) = \bigg(\frac{z + \overline{z}}{2}\bigg) + \bigg(\frac{z - \overline{z}}{2}\bigg) = r \cos(\theta) + r \sin(\theta) ~ i = r e^{i \theta}. \,$

The real numbers are the subset of the complex numbers having $\scriptstyle \Im(z) \,=\, 0\,$ ; likewise the imaginary numbers are the subset with $\scriptstyle \Re(z) \,=\, 0\,$ . The additive identity 0 is both real and imaginary.

Complex arithmetic

The addition and subtraction rules for complex numbers are

$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i \,$

$r_1 \, e^{i \theta_1} \,\pm\, r_2 \, e^{i \theta_2} = [r_1 \cos(\theta_1) \,+\, r_1 \sin(\theta_1) \, i] \,\pm\, [r_2 \cos(\theta_2) \,+\, r_2 \sin(\theta_2) \, i] = [r_1 \cos(\theta_1) \,\pm\, r_2 \cos(\theta_2)] \,+\, [r_1 \sin(\theta_1) \,\pm\, r_2 \sin(\theta_2)] \, i \,$

$= \sqrt{[r_1 \cos(\theta_1) \,\pm\, r_2 \cos(\theta_2)]^2 \,+\, [r_1 \sin(\theta_1) \,\pm\, r_2 \sin(\theta_2)]^2} ~ e^{i \, \arctan\Big[\frac{r_1 \sin(\theta_1) \,\pm\, r_2 \sin(\theta_2)}{r_1 \cos(\theta_1) \,\pm\, r_2 \cos(\theta_2)}\Big]} \,$

The multiplication rule for complex numbers is

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i \,$

$(r_1 \, e^{i \theta_1})(r_2 \, e^{i \theta_2}) = r_1 r_2 \, e^{i (\theta_1 + \theta_2)} = r_1 r_2 \, e^{i ((\theta_1 + \theta_2) \mod (2 \pi))} \,$

The reciprocal of a complex number is

$\frac{1}{z} = \frac{\overline{z}}{z \overline{z}} = \frac{\overline{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2} \,$

$\frac{1}{z} = \frac{1}{r \, e^{i \theta}} = \frac{1}{r} \, e^{-i \theta} = \frac{1}{r} \, e^{i (-\theta \mod (2 \pi))} \,$

The division rule for complex numbers is then

$\frac{a + bi}{c + di} = (a + bi) \frac{1}{c + di} = (a + bi) \frac{c - di}{c^2 + d^2} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \bigg(\frac{ac + bd}{c^2 + d^2}\bigg) + \bigg(\frac{bc - ad}{c^2 + d^2}\bigg)i \,$

$\frac{r_1 \, e^{i \theta_1}}{r_2 \, e^{i \theta_2}} = \frac{r_1}{r_2} \, e^{i (\theta_1 - \theta_2)} = \frac{r_1}{r_2} \, e^{i ((\theta_1 - \theta_2) \mod (2 \pi))} \,$

Complex units and identity elements

There are four units in $\scriptstyle \mathbb{C}\,$ , $\scriptstyle i\,$ being a generator of the multiplicative group $\scriptstyle (\{1, i, -1, -i\}, \cdot)\,$ of order 4:

$i^0 = 1, i^1 = e^{\frac{i \pi}{2}} = i, i^2 = e^{i \pi} = -1, i^3 = e^{\frac{i 3 \pi}{2}} = -i \,$ .

The set of complex numbers together with addition and multiplication is a field with additive identity 0 and multiplicative identity 1.

The effect of the complex units as addends is easily guessed: an increment or decrement of the appropriate real or imaginary part. As multiplicands, the complex units have more varied effect. For the examples in the following section, we'll use $\scriptstyle \frac{1}{2} - 14i \,$ (which is in a way close to a famous complex number) as the other multiplicand:

Multiplication by 1 leaves the real and imaginary parts exactly the same, in value and in sign. Thus, $\scriptstyle (\frac{1}{2} - 14i) \times 1 = \frac{1}{2} - 14i \,$ .
Multiplication by $\scriptstyle i \,$ causes the real and imaginary parts to trade places, and the sign of the new real part is opposite the sign of the old imaginary part. Thus, $\scriptstyle (\frac{1}{2} - 14i) \times i = 14 + \frac{1}{2}i \,$ .
Multiplication by $- 1$ toggles the sign of the real part and toggles the sign of the imaginary part. Thus, $\scriptstyle (\frac{1}{2} - 14i) \times (-1) = -\frac{1}{2} + 14i \,$ .
Multiplication by $\scriptstyle -i \,$ causes the real and imaginary parts to trade places, and the sign of the new imaginary part is opposite the sign of the old real part. Thus, $\scriptstyle (\frac{1}{2} - 14i) \times (-i) = -14 - \frac{1}{2}i \,$ .

The sign of a part is of course moot if that part happens to be 0. So we have multiplication of purely real numbers exactly the same in the complex plane as on the real number line, while multiplication of purely imaginary positive numbers gives purely real negative numbers.

From Euler's identity $\scriptstyle e^{i \pi} \,=\, -1 \,$ we can derive the identities $\scriptstyle e^{i 0} \,=\, 1 \,$ , $\scriptstyle e^{\frac{i \pi}{2}} \,=\, i \,$ and $\scriptstyle e^{\frac{i 3 \pi}{2}} \,=\, -i \,$ , enabling us to restate the above recital of multiplication by units in terms of movement through the complex plane thus:

Multiplication by $\scriptstyle e^{0i} \,$ rotates counterclockwise by an angle of $\scriptstyle 0 \,$ , i.e. null rotation.
Multiplication by $\scriptstyle e^{\frac{\pi i}{2}} \,$ rotates counterclockwise by an angle of $\scriptstyle \frac{\pi}{2} \,$ .
Multiplication by $\scriptstyle e^{\pi i} \,$ rotates counterclockwise by an angle of $\scriptstyle \pi \,$ .
Multiplication by $\scriptstyle e^{\frac{3 \pi i}{2}} \,$ rotates counterclockwise by an angle of $\scriptstyle \frac{3 \pi}{2} \,$ .